Faculty Research Seminar

Abstract: Attempts to prove size lower-bounds for circuits computing
specific functions has been successful establishing bridges from
complexity theory to other areas of computer science and mathematics.
There are several concrete algebraic and combinatorial problems
whose solutions imply circuit lower bounds in different settings.
In this talk we will describe two such problems and their connections to
lower bounds and discuss our results on them:
(1) Matrix Rigidity : Construct an explicit family of (nxn) matrices which
has the following (rigidity) property: in order to bring the rank of the
matrix to below a value r (say n/2) one has to change at least
super-linear (in n) number of entries in the matrix.
(2) Polynomial Identity Testing : Give a efficient deterministic algorithm
for testing whether a given polynomial is identically zero.
We will focus more on the first problem, and describe an algebraically
explicit construction of rigid matrices. We will also state our
results about the second problem when the polynomial is represented by
restricted arithmetic circuits.